Robust Long-Distance Entanglement and a Loophole-Free Bell Test with Ions and Photons

Two trapped ions that are kilometers apart can be entangled by the joint detection of two photons, each coming from one of the ions, in a basis of entangled states. Such a detection is possible with linear optical elements. The use of two-photon interference allows entanglement distribution free of interferometric sensitivity to the path length of the photons. The present method of creating entangled ions also opens up the possibility of a loophole-free test of Bell’s inequalities.

Figure 1

Partial Bell state analyzer as implemented in Ref. 8. One photon comes in through mode $a$, the other through mode $b$. Only if the two photons are in the antisymmetric Bell state $|{\psi }_{-}⟩$ of Eq. (2), there will be one photon in each output mode of the first beam splitter (BS), $c$ and $d$. Therefore a coincidence detection between $D_1$ and $D_3$ or $D_2$ and $D_4$ identifies this state. This was first pointed out in Ref. 7. If the photons are in the state $|{\psi }_{+}⟩$, they both go into $c$ or both into $d$ and are then split by the subsequent polarizing beam splitters (PBS), because they have orthogonal polarizations. Therefore coincidences between $D_1$ and $D_2$ or between $D_3$ and $D_4$ signify a state $|{\psi }_{+}⟩$. For the two other Bell states $|{\varphi }_{+}⟩$ and $|{\varphi }_{-}⟩$ both photons go to the same detector.

Figure 2

Diagram illustrating the timing constraints for a loophole-free Bell experiment.

Figure 3

Relevant levels of ${}^{40}\mathrm{C}\mathrm{a}^{+}$ ions.